1. Field of Invention
This invention relates to cartography, specifically to the relationship between globes and maps.
2. Background of the Invention
The words “globe” and “map” as used herein refer, respectively, to three-dimensional and two-dimensional representations of the surface features of a three-dimensional object such as Earth. Other specific terms herein are used as defined in standard mathematical texts and in Webster's Encyclopedic Unabridged Dictionary of the English Language (1996 Gramercy Books/Random House).
Historically, Earth and the celestial sphere have been the principal objects of interest in mapping. More recently, maps and globes representing the Moon, Mars, and other planets have also been developed. Globes and maps of Earth, in particular, are produced and used in large quantities.
A fundamental problem in cartography is to provide useful maps of solid objects. A useful map portrays with reasonable accuracy the absolute and relative positions of notable features of the original object, as well as distances, directions, and the outline shapes and sizes of notable features. Various methods of geometric projection have been devised to illustrate the features of spherical Earth, for example, on maps. But no map can portray a round surface such as that of Earth with absolute accuracy. Every method of projection produces certain inherent inaccuracies on a map. Representing any solid object on a map always requires a compromise between accuracy and other practical considerations. The representation on a map of any large portion of such an object usually results in some large inaccuracies.
Spherical globes offer the most realistic representations of spherical objects such as Earth, but globes are relatively difficult and costly to manufacture, they tend to be bulky, and they can be awkward to handle, use, and store. Globes can be accurately made, but in use it is difficult to measure or mark distances and directions on a round globe with ordinary tools. Maps are easier and more economical to manufacture than globes of comparable scale and accuracy, and maps are easier to use and store than globes. Maps are therefore useful and popular in spite of their inherent inaccuracies.
3. Prior Art
Fuller's U.S. Pat. No. 2,393,676 (1946) describes the use of the equilateral cuboctahedron as the basis of a useful mapping system. The equilateral cuboctahedron is a symmetrical solid with fourteen flat faces, six of which are squares and eight of which are equilateral triangles. All of the edges of this polyhedron are equal. In Fuller's method each face of the polyhedron becomes a map of a corresponding portion of Earth's surface, with greater overall accuracy than previous projection methods. A particular benefit of Fuller's technique is that the unavoidable inaccuracies are everywhere distributed within the individual faces of the polyhedron, rather than accumulating to produce large errors in a few places, or being concentrated near the edges of the map, as with most projections. This technique offers greater overall accuracy of feature shapes, sizes, and positions throughout the map, when compared to earlier projection methods.
The equilateral cuboctahedron is a particularly apt and convenient basis for representing a sphere by this technique, because each edge of the figure subtends a central angle of exactly sixty degrees. But Fuller's method is not at all limited to the cuboctahedron. The method can use most convex geometric solids as bases. Fuller eventually focussed his attention upon the regular icosahedron, with twenty (20) congruent equilateral triangular faces, as the preferred basis for his mapping system. The icosahedron provides greater overall regularity than the cuboctahedron, and the larger number of polyhedron faces and map segments yields even greater local and overall accuracy.